Relationship between pharmacokinetic half-life and pharmacodynamic half-life in effect-time modeling

A pharmacodynamic parameter relating time-dependent changes of the effect with time-dependent changes of concentrations has yet to be developed. In pharmacokinetics, half-lives (T-1/2kin) are used to describe the relation between concentration (C) and time (t). In pharmacodynamics, often the sigmoid E-max model and the Hill equation are used (E = E-max C-H/(EC50H + C-H)) to describe the relation between effect CE) and concentration (C). To describe the correlation between effect (E) and time (t), a pharmacodynamic half-life (T-1/2dyn) could be estimated if the use of the term half-life is not restricted only to log-linear first order processes, To bisect the drug effect a variable time (t(1-2) = t(2)-t(1)) will be required for this nonlinear process. The bisection of the effect (E-2 = 1/2 E-1) is associated with a decrease in concentrations (C-2 = C-1 exp(-0.693 t(1-2)/T-1/2kin)). A mathematical relationship can be derived between pharmacodynamic half-life (T-1/2dyn = t(1-2)) and pharmacokinetic half-life (T-1/2dyn = T-1/2kin (ln (1 + ln(a)/ln(2))/H) with (a = (EC50H + C-1(H))/(EC50H + C-2(H))). For concentrations in the range of the EC50 value with the Hill coefficient (H = 1), the pharmacodynamic half-life will be 1.6 - 2.0 times the kinetic half-life (T-1/2dyn less than or equal to 2.0 T-1/2kin). For high concentrations (C-1 > EC50), the dynamic half-life will grow much longer than the kinetic half-life, consequently the effect of a drug will not increase but it will last longer. The pharmacodynamic half-life turns out to be a specific estimate for the effect time relation, being a concentration-dependent function of the kinetic half-life.